Numerical and experimental investigations of pseudo-shock systems in a planar nozzle: impact of bypass mass flow due to narrow gaps

Abstract

During previous investigations on pseudo-shock systems, we have observed reproducible differences between measurement and simulations for the pressure distribution as well as for size and shape of the pseudo-shock system. A systematic analysis of the deviations leads to the conclusion that small gaps of $\Delta z=O(10^{-4})$ m between quartz glass side walls and metal contour of the test section are responsible for this mismatch. This paper describes a targeted experimental and numerical study of the bypass mass flow within these gaps and its interaction with the main flow. In detail, we analyze how the pressure distribution within the channel as well as the size, shape and oscillation of the pseudo-shock system are affected by the gap size. Numerical simulations are performed to display the flow inside the gaps and to reproduce and explain the experimental results. Numerical and experimental schlieren images of the pseudo-shock system are in good agreement and show that especially the structure of the primary shock is significantly altered by the presence of small gaps. Extensive unsteady flow simulations of the geometry with gaps reveal that the shear layer between subsonic gap flow and supersonic core flow is subject to a Kelvin–Helmholtz instability resulting in small pressure fluctuations. This leads to a shock oscillation with a frequency of $f= O(10^5) \hbox {s}^{-1}$. The corresponding time scale $\tau$ (s) is 16 times higher than the characteristic time scale $\tau _\delta =\delta /U_\infty$ of the boundary layer given by the ratio of the boundary layer thickness $\delta$ directly ahead of the shock and the undisturbed free stream velocity $U_\infty$. To assess the reliability of our numerical investigations, the paper includes a grid study as well as an extensive comparison of several RANS turbulence models and their impact on the predicted shape of pseudo-shock systems.

Appendices

Appendix A: Geometry

The geometry of the calculated domain is given in Table 1. Between the sections C–D, D–E, G–H and H–I are fillets of $r$ = 200 mm. The corner of the rectangular cross section of H and I is rounded by a fillet. The domain ends at $x$ = 865.56 mm with a fillet of $r$ = 5 mm and a cross section of $A=464.657 \hbox {mm}^2$.

Table 1 Geometry of the calculated domain

The critical cross section B–C is prescribed by (3)

$$\begin{aligned} y(x)=l_{\mathrm{ref}} \times \sum \limits _{i=0}^6 c_i\left( \frac{x}{l_{\mathrm{ref}}}+22.93\right) ^i \end{aligned}$$

(3)

with the following constants:

$$\begin{aligned}&x = \hbox {distance from critical cross section (mm)},\\&l_{\mathrm{ref}} = 1\, \hbox {mm}, \\&c _0 = +5.68224916647265,\\&c_1 = -1.77632698070846 \times 10^{-1},\\&c_2 = +0 \\&c_3 = +1.52798578740639 \times 10^{-4},\\&c_4 = -8.94031304460155 \times 10^{-6},\\&c_5 = +5.45928945559081 \times 10^{-7},\\&c_6 = -1.02048246477148 \times 10^{-8}. \end{aligned}$$

Appendix B: Wall temperature

According to the measurements, the wall temperature in the simulation is prescribed by (4) ($x$ in (mm))

$$\begin{aligned} T_{\mathrm{wall}}(x)=T_{\mathrm{ref}} \times \sum \limits _{i=0}^{6} d_i\left( \frac{x}{l_{\mathrm{ref}}}\right) ^{i} \end{aligned}$$

(4)

with the following constants:

$$\begin{aligned}&x = \hbox {distance from critical cross section (mm)},\\&l_{\mathrm{ref}} = 1\, \hbox {mm}, \\&T_{\mathrm{ref}} = 1 K, \\&d_0 = +2.9332 \times 10^{2}, \\&d_1 = -9.1309 \times 10^{-02},\\&d_2 = +6.5255 \times 10^{-04},\\&d_3 = -1.5826 \times 10^{-06},\\&d_4 = +1.7119 \times 10^{-09},\\&d_5 = -7.6983 \times 10^{-13},\\&d_6 = +7.6754 \times 10^{-17}. \end{aligned}$$